Online Job Assignment

Abstract

Primarily motivated by applications in cloud computing marketplaces, we study a simple, yet powerful, adversarial online allocation problem in which jobs of various lengths arrive online over continuous time and upon arrival are immediately and irrevocably assigned to one of the available offline servers (if any). The servers have given initial capacity. Once a job is assigned to a server, the job uses one unit of capacity for its duration and releases this unit once it is finished. In addition, the algorithm obtains a reward from this assignment. We consider a general heterogeneous model where both the reward and duration of each job assignment can depend on the job-server pair. The goal of the online algorithm is to sequentially assign jobs to servers to maximize the total collected rewards and aims to be online competitive against an omniscient optimal benchmark that knows all the jobs in advance. Our main result is designing a new family of online assignment algorithms, which we call "Forward-Looking BALANCE (FLB)", and using a primal-dual framework to show that this family obtains an asymptotically optimal competitive ratio in various regimes if we choose a proper member.

In summary, this meta-algorithm has two important primitives: (i) keeping track of the capacity used for each server at each time and applying a penalty function to this quantity, and (ii) adjusting the reward of assigning an arriving job to a server by subtracting the total penalty of a particularly chosen subset of future times (referred to as inspection times). The FLB algorithm then assigns the arriving job to the server with the maximum adjusted reward. In the general setting, if R and D are the ratios of maximum over minimum rewards and durations, we show that there exists a choice of these primitives so that the FLB algorithm obtains a competitive ratio of log(RD)+3loglog(max(R,D))+O(1) as the initial capacities converge to infinity. Furthermore, in the special case of the fixed reward setting (R=1), we show that the FLB algorithm with a different choice of primitives obtains a near-optimal asymptotic competitive ratio of log(D)+O(1) (with different constants in different settings). Our main analysis combines a dual-fitting technique that takes advantage of the configuration LP benchmark for this problem and a novel inductive argument to establish our results, which might be of independent interest.

Keywords: Online resource allocation, online matching, cloud computing, job scheduling, online algorithms, primal-dual